Fundamental structures of algebra and discrete mathematics - Foldes, Stephan (Wiley, 1994)
Structures are defined by laws of composition, rules of generation, and relations. The objects on which these laws operate may be numbers, geometric objects like points and lines, or abstract symbols. Algebra is the study of mathematical laws, with a search for general principles that do not depend on what the objects are. Fundamental Structures of Algebra and Discrete Mathematics is an introduction to the twelve basic kinds of structures - sets, ordered sets, groups, rings, fields, vector spaces, graphs, lattices, matroids, topological spaces, universal algebras, and categories - that underlie algebra and discrete mathematics. Beginning with the most basic type of structure, sets, this unique reference provides a detailed look at the theoretical underpinning of each structure, shedding light on the significance of each structure as well as their interrelation. Using a self-contained approach that requires little previous knowledge of mathematical definitions, results, or methods, the book examines selected key aspects of these structures, including closure systems, generators, substructures, homomorphisms and congruences, equational axioms, connections with basic set theory, finiteness conditions, combinatorial properties, and discrete algorithmic procedures. Several classical results are proved, including the Abel-Ruffini theorem on unsolvability by radicals, Helly's theorem on intersecting convex sets, and a simplified version of the Godel-Herbrand completeness theorem. Many of the results are relevant to current research. Highly interactive in approach, the book features numerous exercises and examples woven throughout the text that allow the reader to become fully acquainted with the character and function of each structure. Additional questions, designed to stretch the reader's analytical skills, appear at the end of each section. Similar to a grammar book that explains the fundamental structures essential to mastering a language, Fundamental Structures of Algebra and Discrete Mathematics is a systematic examination of the basic algebraic structures needed to understand and manipulate advanced mathematical concepts. A cornerstone reference that is both a clear primer and rigorous study guide, Fundamental Structures of Algebra and Discrete Mathematics is indispensable to the student and professional seeking to learn or use the methods of modern algebra.